Espectro do operador Laplaciano de Dirichlet em tubos deformados
Abstract
Let Ω be a deformed tube in R3 and −∆Ω
D the Dirichlet Laplacian operator
in Ω. In this work, we are going to study the spectrum σ(−∆Ω
D) of the operator
−∆Ω
D. More precisely, we are going to analize how the geometrical characteris-
tics of Ω can influence in the set σ(−∆Ω
D). Firstly, we are going to show that,
under certain conditions, the essential spectrum σess(−∆Ω
D) of −∆Ω
D is the same,
independent if the tube is straight, curved or twisted. In regard to the discrete
spectrum σdis(−∆Ω
D) of −∆Ω
D, we are going to show that if Ω is a curved tube,
then σdis(−∆Ω
D) is a non empty set. Furthermore, if Ω is a twisted tube, then
σdis(−∆Ω
D) is a empty set. In the case where Ω is lightly curved and twisted
simultaneously, we are going to see that the discrete spectrum remains empty.